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Dec 16, 2017
12/17

by
Daniel Pollack

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Tracklist: 1. Liszt: Consolation No.3 in D flat Major 2. Ravel: Pavane Pour Une Infante Defunte 3. Chopin: Nocturne Op.27, No.2 in D flat Major 4. Debussy: Girl With The Flaxen Hair 5. Liszt-Siloti: Un Sospiro (A Sigh) Concert Etude in D flat Major 6. Chopin: Nocturne No. 20, Op. Posthumous in C sharp minor 7. Schumann: Carnaval Op.9 Chiarina & Chopin 8. Debussy: Arabesque No.1 9. Brahms: Intermezzo Op.118, No.2 in A Major 10. Scriabin: Etude Op.2, No.1 in C sharp minor 11. Rachmaninov:...

Source: CD

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Sep 18, 2013
09/13

by
Rafe Mazzeo; Daniel Pollack

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In this paper we survey a number of recent results concerning the existence and moduli spaces of solutions of various geometric problems on noncompact manifolds. The three problems which we discuss in detail are: I. Complete properly immersed minimal surfaces in $\RR^3$ with finite total curvature. II. Complete embedded surfaces of constant mean curvature in $\RR^3$ with finite topology. III. Complete conformal metrics of constant positive scalar curvature on $M^n \setminus \Lambda$, where...

Source: http://arxiv.org/abs/dg-ga/9601008v1

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Sep 18, 2013
09/13

by
Justin Corvino; Daniel Pollack

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We survey some results on scalar curvature and properties of solutions to the Einstein constraint equations. Topics include an extended discussion of asymptotically flat solutions to the constraint equations, including recent results on the geometry of the center of mass of such solutions. We also review methods to construct solutions to the constraint equations, including the conformal method, as well as gluing techniques for which it is important to understand both conformal and non-conformal...

Source: http://arxiv.org/abs/1102.5050v1

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Sep 23, 2013
09/13

by
Piotr T. Chrusciel; Daniel Pollack

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We construct large families of initial data sets for the vacuum Einstein equations with positive cosmological constant which contain exactly Delaunay ends; these are non-trivial initial data sets which coincide with those for the Kottler-Schwarzschild-de Sitter metrics in regions of infinite extent. From the purely Riemannian geometric point of view, this produces complete, constant positive scalar curvature metrics with exact Delaunay ends which are not globally Delaunay. The ends can be used...

Source: http://arxiv.org/abs/0710.3365v2

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Sep 22, 2013
09/13

by
Piotr T. Chrusciel; James Isenberg; Daniel Pollack

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We establish an optimal gluing construction for general relativistic initial data sets. The construction is optimal in two distinct ways. First, it applies to generic initial data sets and the required (generically satisfied) hypotheses are geometrically and physically natural. Secondly, the construction is completely local in the sense that the initial data is left unaltered on the complement of arbitrarily small neighborhoods of the points about which the gluing takes place. Using this...

Source: http://arxiv.org/abs/gr-qc/0409047v1

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Sep 19, 2013
09/13

by
Yvonne Choquet-Bruhat; James Isenberg; Daniel Pollack

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We study the constraint equations for the Einstein-scalar field system on compact manifolds. Using the conformal method we reformulate these equations as a determined system of nonlinear partial differential equations. By introducing a new conformal invariant, which is sensitive to the presence of the initial data for the scalar field, we are able to divide the set of free conformal data into subclasses depending on the possible signs for the coefficients of terms in the resulting...

Source: http://arxiv.org/abs/gr-qc/0610045v2

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Sep 22, 2013
09/13

by
Emmanuel Hebey; Frank Pacard; Daniel Pollack

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We establish new existence and non-existence results for positive solutions of the Einstein-scalar field Lichnerowicz equation on compact manifolds. This equation arises from the Hamiltonian constraint equation for the Einstein-scalar field system in general relativity. Our analysis introduces variational techniques, in the form of the mountain pass lemma, to the analysis of the Hamiltonian constraint equation, which has been previously studied by other methods.

Source: http://arxiv.org/abs/gr-qc/0702031v1

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Sep 22, 2013
09/13

by
Piotr T. Chrusciel; James Isenberg; Daniel Pollack

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We present a local gluing construction for general relativistic initial data sets. The method applies to generic initial data, in a sense which is made precise. In particular the trace of the extrinsic curvature is not assumed to be constant near the gluing points, which was the case for previous such constructions. No global conditions on the initial data sets such as compactness, completeness, or asymptotic conditions are imposed. As an application, we prove existence of spatially compact,...

Source: http://arxiv.org/abs/gr-qc/0403066v2

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Sep 24, 2013
09/13

by
Rob Kusner; Rafe Mazzeo; Daniel Pollack

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We examine the space of surfaces in $\RR^{3}$ which are complete, properly embedded and have nonzero constant mean curvature. These surfaces are noncompact provided we exclude the case of the round sphere. We prove that the space $\Mk$ of all such surfaces with $k$ ends (where surfaces are identified if they differ by an isometry of $\RR^{3}$) is locally a real analytic variety. When the linearization of the quasilinear elliptic equation specifying mean curvature equal to one has no...

Source: http://arxiv.org/abs/dg-ga/9408004v1

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Sep 22, 2013
09/13

by
James Isenberg; David Maxwell; Daniel Pollack

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We extend the conformal gluing construction of Isenberg-Mazzeo-Pollack [18] by establishing an analogous gluing result for field theories obtained by minimally coupling Einstein's gravitational theory with matter fields. We treat classical fields such as perfect fluids and the Yang-Mills equations as well as the Einstein-Vlasov system, which is an important example coming from kinetic theory. In carrying out these extensions, we extend the conformal gluing technique to higher dimensions and...

Source: http://arxiv.org/abs/gr-qc/0501083v1

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Sep 19, 2013
09/13

by
James Isenberg; Rafe Mazzeo; Daniel Pollack

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We prove that there are no restrictions on the spatial topology of asymptotically flat solutions of the vacuum Einstein equations in (n+1)-dimensions. We do this by gluing a solution of the vacuum constraint equations on an arbitrary compact manifold to an asymptotically Euclidean solution of the constraints on R^n. For any compact manifold which does not admit a metric of positive scalar curvature, this provides for the existence of asymptotically flat vacuum spacetimes with no maximal slices....

Source: http://arxiv.org/abs/gr-qc/0206034v2

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Sep 20, 2013
09/13

by
Yvonne Choquet-Bruhat; James Isenberg; Daniel Pollack

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We use the conformal method to obtain solutions of the Einstein-scalar field gravitational constraint equations. Handling scalar fields is a bit more challenging than handling matter fields such as fluids, Maxwell fields or Yang-Mills fields, because the scalar field introduces three extra terms into the Lichnerowicz equation, rather than just one. Our proofs are constructive and allow for arbitrary dimension (>2) as well as low regularity initial data.

Source: http://arxiv.org/abs/gr-qc/0506101v1

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Sep 23, 2013
09/13

by
James Isenberg; Rafe Mazzeo; Daniel Pollack

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We establish a general gluing theorem for constant mean curvature solutions of the vacuum Einstein constraint equations. This allows one to take connected sums of solutions or to glue a handle (wormhole) onto any given solution. Away from this handle region, the initial data sets we produce can be made as close as desired to the original initial data sets. These constructions can be made either when the initial manifold is compact or asymptotically Euclidean or asymptotically hyperbolic, with...

Source: http://arxiv.org/abs/gr-qc/0109045v3

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Sep 18, 2013
09/13

by
Yvonne Choquet-Bruhat; James Isenberg; Daniel Pollack

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The Einstein-scalar field theory can be used to model gravitational physics with scalar field matter sources. We discuss the initial value formulation of this field theory, and show that the ideas of Leray can be used to show that the Einstein-scalar field system of partial differential equations is well-posed as an evolutionary system. We also show that one can generate solutions of the Einstein-scalar field constraint equations using conformal methods.

Source: http://arxiv.org/abs/gr-qc/0611009v1

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Sep 23, 2013
09/13

by
Rafe Mazzeo; Daniel Pollack; Karen Uhlenbeck

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Complete, conformally flat metrics of constant positive scalar curvature on the complement of $k$ points in the $n$-sphere, $k \ge 2$, $n \ge 3$, were constructed by R\. Schoen [S2]. We consider the problem of determining the moduli space of all such metrics. All such metrics are asymptotically periodic, and we develop the linear analysis necessary to understand the nonlinear problem. This includes a Fredholm theory and asymptotic regularity theory for the Laplacian on asymptotically periodic...

Source: http://arxiv.org/abs/dg-ga/9406004v1

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Sep 18, 2013
09/13

by
Piotr T. Chrusciel; Frank Pacard; Daniel Pollack

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We present a gluing construction which adds, via a localized deformation, exactly Delaunay ends to generic metrics with constant positive scalar curvature. This provides time-symmetric initial data sets for the vacuum Einstein equations with positive cosmological constant with exactly Kottler-Schwarzschild-de Sitter ends, extending the results in arXiv:0710.3365 [gr-qc].

Source: http://arxiv.org/abs/0803.1817v1

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Sep 21, 2013
09/13

by
Rafe Mazzeo; Frank Pacard; Daniel Pollack

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We establish a general `gluing theorem', which states roughly that if two nondegenerate constant mean curvature surfaces are juxtaposed, so that their tangent planes are parallel and very close to one another, but oppositely oriented, then there is a new constant mean curvature surface quite near to this configuration (in the Hausdorff topology), but which is a topological connected sum of the two surfaces. Here nondegeneracy refers to the invertibility of the linearized mean curvature...

Source: http://arxiv.org/abs/math/9905077v1

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Sep 21, 2013
09/13

by
Michael Eichmair; Gregory J. Galloway; Daniel Pollack

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We introduce a natural generalization of marginally outer trapped surfaces, called immersed marginally outer trapped surfaces, and prove that three dimensional asymptotically flat initial data sets either contain such surfaces or are diffeomorphic to R^3. We establish a generalization of the Penrose singularity theorem which shows that the presence of an immersed marginally outer trapped surface generically implies the null geodesic incompleteness of any spacetime that satisfies the null energy...

Source: http://arxiv.org/abs/1204.0278v2

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Sep 23, 2013
09/13

by
Rafe Mazzeo; Daniel Pollack; Karen Uhlenbeck

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We give a general procedure for gluing together possibly noncompact manifolds of constant scalar curvature which satisfy an extra nondegeneracy hypothesis. Our aim is to provide a simple paradigm for making `analytic' connected sums. In particular, we can easily construct complete metrics of constant positive scalar curvature on the complement of certain configurations of an even number of points on the sphere, which is a special case of Schoen's \cite{S1} well-known, difficult construction....

Source: http://arxiv.org/abs/dg-ga/9511018v1

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Jun 28, 2018
06/18

by
Lars Andersson; Mattias Dahl; Gregory J. Galloway; Daniel Pollack

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We study the relationship between initial data sets with horizons and the existence of metrics of positive scalar curvature. We define a Cauchy Domain of Outer Communications (CDOC) to be an asymptotically flat initial set $(M, g, K)$ such that the boundary $\partial M$ of $M$ is a collection of Marginally Outer (or Inner) Trapped Surfaces (MOTSs and/or MITSs) and such that $M\setminus \partial M$ contains no MOTSs or MITSs. This definition is meant to capture, on the level of the initial data...

Topics: Differential Geometry, General Relativity and Quantum Cosmology, Mathematics

Source: http://arxiv.org/abs/1508.01896

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Sep 22, 2013
09/13

by
Piotr T. Chruściel; Gregory J. Galloway; Daniel Pollack

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We provide an introduction to selected recent advances in the mathematical understanding of Einstein's theory of gravitation.

Source: http://arxiv.org/abs/1004.1016v2